关于Herglotz的积分表示与三角矩问题

Japanese journal of mathematics :transactions and abstracts Pub Date : 1900-01-01 DOI:10.4099/JJM1924.29.0_35

Yūsaku Komatu

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引用次数: 7

摘要

我们可以进一步假设ƒÏ(ƒÕ)-Rƒ3(0)ƒÕ/2ƒÎ在其最初定义的区间之外周期性地延续，周期为2ƒÎ。在课堂上的几个问题的研究表明，这个表示公式往往提供了一个强大的工具;例如，参见[8]。为了以后参考，我们在这里重述一下它的推导大纲。首先将泊松积分表示应用于函数ƒ3(z)•_?? ?_ regular for |z|•…R <1，我们得到

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On the integral representation of Herglotz and the trigonometric moment problem

We may further suppose that ƒÏ(ƒÕ)-Rƒ3(0)ƒÕ/2ƒÎ is continued periodically beyond its original interval of definition with the period 2ƒÎ. In studying several problems on the class, it has been indicated that this re presentation formula often offers a powerful tool; cf., for instance, [8]. For the sake of later reference, we reproduce here the outline of its derivation. Applying first the Poisson integral representation to a function ƒ3(z)• ̧_??_ regular for |z|•...r<1, we get

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Japanese journal of mathematics :transactions and abstracts

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